I have been looking at radial basis function interpolation: $f(x) = \sum w_i \phi_i(||x-x_i||)$ and examining the different kernels e.g. $\phi(r) = e^{-(\epsilon r) ^2}$ which are generally maximised when $r = 0$ i.e. so the weights assigned to the training data which are close to $x$ contribute the most to its value. However thin plate spline kernels $\phi(r) = r^2 log(r)$ grow after $r$ becomes large enough. Intuitively this doesn't make sense to me - surely it is dangerous to have growing influence of the data points far away from $x$?
I understand the TPS is designed to be resistant to bending etc. so it makes sense that we include strong influence from points further - but intuitively it still seems contradictory. Are there any other reasons/situations where this would be used, and why we would want a kernel function that increases with $r$?
